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MHT-CET : Mathematics Entrance Exam

MHT - CET : Mathematics - Parabola Formulae Page 1

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Parabola

 

Def: Parabola is the locus of a point on a plane, which is equidistant from a fixed point and a fixed line of that plane.
If
P is the variable point, S is the fixed point and PM is the distance from a fixed line 'd', then PS = PM (Focus-Directrix property)

Standard form:
y2 = 4ax, y2 = - 4ax, x2 = 4ay, x2 = - 4ay
Important information:

 

 

 

y2 = 4ax

y2 = - 4ax

x2 = 4ay

x2 = - 4ay

Co-ordinate of Vertex

(0, 0)

(0, 0)

(0, 0)

(0, 0)

Co-ordinate of focus

(a, 0)

(- a, 0)

(0, a)

(0, - a)

Axis

x-axis

x-axis

y-axis

y-axis

Equation of directrix

x + a = 0

x - a = 0

y + a = 0

y - a = 0

Length of Latus rectum

4a

4a

4a

4a

Co-ordinate of the end of Latus rectum

(a, 2a)

(a, - 2a)

(- a, 2a)

(- a, - 2a)

(2a, a)

(- 2a, a)

(2a, - a)

(- 2a, - a)

Parametric Co-ordinate

x = at2, y =
2
at

x = - at2, y = 2at

x = 2at , Y = at2

x = 2at, y = - at2

Equation of axis

y = 0

y = 0

x = 0

x = 0

Extent(Quadrants)

I & IV

II & III

I & II

III & IV

 

 

General Equation: (y - b)2 = 4a(x - a)

 

 

 

 

 

Important information:

 

 

Co-ordinate of Vertex

Co-ordinate of focus

Equation of directrix

Length of Latus rectum

Co-ordinate of the end of Latus rectum

Equation of axis

Parametric Co

(a, b)

(a + a,b)

X = a + a

4a

(a + a, 2a + b) (a + a, - 2a + b)

y = b

x = at2 + a,

y = 2at + b

 

 

 

 

 

Tangent to a parabola:

 

 

Equation of tangent to a parabola y2 = 4ax at (x1, y1) is yy1 = 2a(x + x1)
Equation of tangent to a parabola
y2 = 4ax at (at2,2at) is yt = x + at2.

 

 

General equation of tangent to a parabola y2 = 4ax in terms of slope is y = mx

a

m

 

 

General equation of tangent to y2 = 4ax in terms of parameter 't' is yt = x + at2.

 

 

Condition of tangency c =  

a

m

 

 

Point of contact is

(

 - 

 

a

m2

,

a

m

)

 

 

 

 

 

Ellipse:

 

Def: Ellipse is the locus of a point on a plane such that the ratio e of its distances from a fixed point and a fixed line of that plane is constant where e <1.
If
P is the variable point, S is the fixed point and PM is the distance from a fixed line 'd' then PS = e PM (Focus-Directrix property)

 

 

 

 

Standard form: 

x2

a2

  +  

 

y2

b2

 = 1(a > b)

 

 

 

 

 

x2

a2

  +  

 

y2

b2

 = 1(a > b)

 

Co-ordinate of Vertex

(0, 0)

Co-ordinate of focus

( ae, 0)

Major axis

x-axis

Length of Major axis

2a

Minor axis

y-axis

Length of Minor axis

2b

Equation of directrix

x

a

e

 = 0

Length of Latus rectum

2b2

a

Co-ordinate of the end of Latus rectum

(

 ae,

b2

a

)

Relation between a, b and e

b2 = a2(1 - e2)

Parametric Co-ordinate

x = a cosq, y = b sinq

 

 

 

 

 

Tangent to an Ellipse:

 

 

Equation of tangent to an ellipse  

x2

a2

 + 

y2

b2

 = 1 (a > b) at (x1, y1) is

xx1

a2

 + 

yy1

b2

 = 1

 

Equation of tangent to an ellipse  

x2

a2

 + 

y2

b2

 = 1 at (a cosq, b sinq) is

xcosq

a

 + 

ysinq

b

 = 1

 

General equation of tangent to an ellipse  

x2

a2

 +

y2

b2

= 1 in terms of slope m is

 

   y = mx

 

 

Condition of tangency c

 

 

Point of contact is  

(

m

 

 , 

)

 

 

Hyperbola:

 

Def: Hyperbola is the locus of a point on a plane such that the ratio e of its distances from a fixed point and a fixed line of that plane is constant where e >1.
If
P is the variable point, S is the fixed point and PM is the distance from a fixed line 'd' then PS = e PM (Focus-Directrix property)

Standard form: 

x2

a2

   -   

 

y2

b2

 = 1(a > b)

 

 

 

 

 

Important information:

 

 

x2

a2

   -   

 

y2

b2

 = 1(a > b)

 

Co-ordinate of Centre

(0,0)

Co-ordinate of focus

( ae, 0)

Major axis

x-axis

Length of Major axis

2a

Minor axis

y-axis

Length of Minor axis

2b

Equation of directrix

X

a

e

 = 0

Length of Latus rectum

2b2

a

Relation between a,b and e

b2 = a2(e2 - 1 )

Co-ordinate of the end of Latus rectum

(

 ae,

b2

a

)

Parametric Co-ordinate

X = a secq, y = b tanq

 

 

 

 

 

Tangent to a Hyperbola

 

 

Equation of tangent to a hyperbola 

x2

a2

 - 

y2

b2

 = 1 (a > b) at (x1, y1) is

xx1

a2

 - 

yy1

b2

 = 1

 

Equation of tangent to a hyperbola 

x2

a2

 - 

y2

b2

 = 1 at (a secq, b tanq) is

x secq

a

 + 

y tanq

b

 = 1

 

General equation of tangent to a hyperbola 

x2

a2

 - 

y2

b2

 = 1 in terms of slope m is

 

y = mx

 

 

Condition of tangency c

 

 

Point of contact is  

(

 

m

 

)

 

 

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